A girl must be like a blossomWith honey for just one man.A man must be like honey beeAnd gather all he can.To fly from blossom to blossomA honey bee must be free,But blossom must not ever flyFrom bee to bee to bee.

--King Mongut of Siam to Anna Leonowensin the 1951 Broadway musical, The King and Iby Richard Rogers and Oscar Hammerstein

nasmuch
as blossoms outnumber bees by a lot, the aptness of Mongut's metaphor resides
in mathematics not entomology. In the first place, worker bees are
female. Meanwhile, sophisticated solvers will doubtless answer the
question in
the puzzle as follows:

Which of the two cases would surprise you?

Both.

If your answer is Case 1, you are in good -- at least
plentiful -- company. Case 1 was derived from a survey recently published
by the U.S. Government, which concluded that men reported a median of
seven
female sex partners, while women reported a median of four male
sex partners.

According to an August 12, 2007 article in
The New
York Times, entitled "The Myth, the Math, the Sex" by Gina Kolata,
in many research studies conducted all over the world, males report more
sexual partners than females.

Conventional explanations based on natural
selection will invariably embrace behavior differences consistent with
Case 1. However, in human populations wherein 'blossoms' and 'bees'
are approximately equal in number, both Case 1 and Case 2 are just
about mathematically
impossible, as we shall see.

Explaining the Impossible

The subject of this puzzle is extremely popular among
psychologists and important to researchers on health-care issues.
A web-search will confirm the availability of studies by the hundreds,
each with lopsided results not much different from those on which the Materix
puzzle is based. Explanations vary widely. Here begins our
metanalysis...

Lopsided Population

We start by creating a matrix of a population, labeling
the columns with male identities M1,
M2,
M3,...Mm
and the rows with female identities
F1,
F2,
F3,...
Fn.
Let an entry of 1 in the matrix indicate a romantic relationship
between the corresponding male and female -- a mating (hence the
coinage here of the portmanteau 'materix').
After sprinkling in some 1s, here is how our population
materix
might look...

M1

M2

M3

M4

M5

M6

...

Mm

F1

1

1

F2

1

F3

1

1

1

F4

1

F5

1

1

1

F6

1

...

1

Fn

1

Now, we add up all the columns and divide by m
then add up all the rows and divide by n,
thereby finding the mating averages for m
males and for n
females. If m
= n, of course, the averages
come out the same for both genders -- whatever the pattern of entries.
That rules out both Case 1 and Case 2 and confirms our "both" as
the solution to the Materix puzzle.
Or so it seems.

It is easy to show that for Case 1 to prevail, m
/ n = 4 / 7. and that for Case
2, m / n
= 7 / 4. For that amount of lopsidedness, there would have to be
75% more females than males in the population or vice versa. Give
the previous sentence exclamatory punctuation if you like.

Lopsided Sample

Both the Materix puzzle and
the survey by the U.S. Government postulated...

(a) "questionnaires completed by an equal number
of males and females" and...(b) responders "selected at random from the same population."

In the populationmaterix
above, let us suppose that questionnaires were given only to M1,
M3,
M5,
F2,
F4,
F6.
The averages would come out two for males and one for females,
even though both genders participated equally in what might be called a
sampling
materix.

By the way, you may have noticed that none of the males
and females in that particular samplingmaterix
ever
mated with each other. Whenever that takes place, the entry
of 1 applies to both genders
equally, shifting the averages toward each other.

rothels
are often cited as exceptions, since they trade in transitory romantic
relationships for males while not bumping up the averages for females within
the population. Please note, however, the magnitude of the seven-to-four
ratio requires the existence of a huge romance-for-finance industry, at
the same time excluding prostitutes from the populationmaterix.
Oh, and what about female passengers on cruise ships and gigolos at ports
o' call?

Nevertheless, several reseachers offer prostitution as
an explanation for the lopsidedness. Entries outside the sampling
materix,
whether occurring in bordellos or motel rooms, will indeed have the power
to skew the averages one way or the other. Let us suppose for the moment
that the samplingmaterix does accurately
represent the populationmaterix.
Some people will insist that, if m
= n, the lopsidedness in Case 1 means
that a large number of males must simply be selecting romantic partners
from outside the populationmaterix.
Sure, and females don't ever do that?

Here's the kicker: statistics for partners outside of
the populationmaterix really ought
to be tabulated along with their own romantic experiences, which effectively
appropriates their outside partners into the populationmaterix.
In concept, then, both m
and
n
may
be driven to increase exponentially, limited ultimately by the carrying
capacity of the planet whereupon m
=
n and neither Case 1 nor Case 2 can
prevail.

Lopsided Desires

It may be instructive to note that one study did not inquire
about cumulative life-long mating experiences but about desires
instead. The question was of the form, "How many romantic relationships
do you want to have over the next year?" Differences in the
gender-specific responses were quite pronounced: males saying "many" and
females saying "only one."

Lopsided Orientation

Each romantic relationship being considered on this page
assumes the participation of one gynotaxic
male and one androtaxic
female. Suppose that there is a common flaw herein and also
in the worldwide studies -- that they all inadvertently included same-sex
partnerships. Such romantic relationships would count twice
for androtaxic males and zero for females. Or vice versa --
twice
for gynotaxic females and zero for males. The explanation
for the lopsidedness in Case 1 would then depend on the relative frequencies
of each sexual
orientation in the population compounded with the
lopsided distribution of romantic partners reported by males and females
in each category.

What about age? Curiously, the research to date
appears to neglect this question. Unmistakable gender asymmetry is commonly
observed, with males at any age favoring romantic relationships with females
at younger ages while females at any age being potentially attracted to
males of every age. Let us visualize that the questionnaires are
segregated into stacks according to age.

The study would surely find that young men have fewer
opportunities for romantic relationships than young women. A typical
male’s mating prospects will start out limited mainly to females at his
own age and then increase with age.

A young female, on the other hand, has more abundant choices
than a young male. Beginning at the earliest adult age, a female
has romantic access to the whole population of males, although young chicks
will seldom go for old codgers when there are plenty of young studs around.

ost
of the statistical studies you will find on the web take place in university
settings. An
eligibility materix
limited in scope to young people will surely have m
> n, which favors Case 2 not Case 1.
As males get older, the number of eligible females gets steadily larger
and so, presumably, will the reported quantity of romances with them.
Meanwhile, unlike the male, the female enjoys a target population of eligible
males that will decline steadily with her own age.

An eligibilitymaterix
of middle-aged people will probably have m
< n, which does favor Case 1 --
but only if it's not too late. Matrimony and parenting will put the
statistical brakes on accumulating romantic relationships by both males
and females.

Lopsided Median

Sophisticated solvers may notice that the Materix
puzzle statement used the word "typical" ("typical males"..."typical females").
You are invited to interpret that phraseology as merely an informal way
of saying "average" or mean. Oh, but then it is surely fair
to take "typical" to mean median, as explicitly depicted in the
U.S. Government report.

Some writers have tried to explain gender asymmetry in
terms of the mathematical distinctions that can arise between median
and mean. That subject is treated at length in Revenge
of the DAR, where the power of the median is seen to discard
outliers
(trollops and Lotharios) in estimating "central tendency" -- exactly the
opposite
of what one needs for explaining the lopsidedness in either Case 1 or Case
2.

Lopsided Reporting

One final parsing of the Materix
puzzle statement brings the word "reported" into consideration. And
for good reason. The data in all of the studies necessarily comes
from the subjects themselves, with no feasible corroboration.

Several writers argue that gender-specific standards of
deportment for many cultures will account for the differences in the reported
numbers -- that if asked, a man, believing he will be admired for having
accumulated a lot of romantic "conquests," may have an incentive to exaggerate,
and a woman, expecting to garner more approval for having withheld her
charms, will tend to minimize her numbers.

Some studies blame differences on the value assigned
by the respective participants to past romantic relationships. Women
are said to remember the specifics of each and every relationship, while
men remember only generalities, which allows plenty of fuzziness in estimating
the actual number.

Lopsided Anonymity

One particular study may have teased some statistical
truth out of the reported numbers using anonymity as the
independent variable. The sample-sizes (in the dozens) were too small
to be any more than suggestive; nevertheless, here is the protocol.

Half of the males and half of the females were assured
that their questionnaires would be kept anonymous. The others were
told that the identities of participants would be disclosed at the end
of the study. The results were striking. Almost deserving an
exclamation point.

In the anonymous group, the quantities of reported romantic
relationships were about the same for both males and females --
between six and seven different partners for each volunteer.

On the non-anonymous questionnaires, males reported
about the same quantity of romantic partners -- if anything, higher
than the anonymous group, within the accuracy afforded by the small sample-size.

Meanwhile, the females who were deprived of anonymity, reported
about half the quantities of romantic relationships.

Thus we try to explain objective impossibilities by conjecturing
about subjective possibilities. In the words of King Mongut of Siam,
"There are times I almost think I am not sure of what I absolutely know."